- The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$
The line \(l\) is the normal to \(E\) at the point \(P ( 5 \cos \theta , 3 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\)
- Using calculus, show that an equation for \(l\) is
$$5 x \sin \theta - 3 y \cos \theta = 16 \sin \theta \cos \theta$$
Given that
- \(\quad l\) intersects the \(y\)-axis at the point \(Q\)
- the midpoint of the line segment \(P Q\) is \(M\)
- determine the exact maximum area of triangle \(O M P\) as \(\theta\) varies, where \(O\) is the origin.
You must justify your answer.